Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > gcdmultiplezOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of gcdmultiplez 16127 as of 12-Jan-2024. Extend gcdmultiple 16128 so 𝑁 can be an integer. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
gcdmultiplezOLD | ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd (𝑀 · 𝑁)) = 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7242 | . . . 4 ⊢ (𝑁 = 0 → (𝑀 · 𝑁) = (𝑀 · 0)) | |
2 | 1 | oveq2d 7250 | . . 3 ⊢ (𝑁 = 0 → (𝑀 gcd (𝑀 · 𝑁)) = (𝑀 gcd (𝑀 · 0))) |
3 | 2 | eqeq1d 2741 | . 2 ⊢ (𝑁 = 0 → ((𝑀 gcd (𝑀 · 𝑁)) = 𝑀 ↔ (𝑀 gcd (𝑀 · 0)) = 𝑀)) |
4 | nncn 11867 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ) | |
5 | zcn 12210 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
6 | absmul 14890 | . . . . . . 7 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁))) | |
7 | 4, 5, 6 | syl2an 599 | . . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁))) |
8 | nnre 11866 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ) | |
9 | nnnn0 12126 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0) | |
10 | 9 | nn0ge0d 12182 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 0 ≤ 𝑀) |
11 | 8, 10 | absidd 15018 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (abs‘𝑀) = 𝑀) |
12 | 11 | oveq1d 7249 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → ((abs‘𝑀) · (abs‘𝑁)) = (𝑀 · (abs‘𝑁))) |
13 | 12 | adantr 484 | . . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) · (abs‘𝑁)) = (𝑀 · (abs‘𝑁))) |
14 | 7, 13 | eqtrd 2779 | . . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (abs‘(𝑀 · 𝑁)) = (𝑀 · (abs‘𝑁))) |
15 | 14 | oveq2d 7250 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd (abs‘(𝑀 · 𝑁))) = (𝑀 gcd (𝑀 · (abs‘𝑁)))) |
16 | 15 | adantr 484 | . . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → (𝑀 gcd (abs‘(𝑀 · 𝑁))) = (𝑀 gcd (𝑀 · (abs‘𝑁)))) |
17 | simpll 767 | . . . . 5 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → 𝑀 ∈ ℕ) | |
18 | 17 | nnzd 12310 | . . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → 𝑀 ∈ ℤ) |
19 | nnz 12228 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
20 | zmulcl 12255 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) | |
21 | 19, 20 | sylan 583 | . . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) |
22 | 21 | adantr 484 | . . . 4 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → (𝑀 · 𝑁) ∈ ℤ) |
23 | gcdabs2 16121 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (𝑀 gcd (abs‘(𝑀 · 𝑁))) = (𝑀 gcd (𝑀 · 𝑁))) | |
24 | 18, 22, 23 | syl2anc 587 | . . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → (𝑀 gcd (abs‘(𝑀 · 𝑁))) = (𝑀 gcd (𝑀 · 𝑁))) |
25 | nnabscl 14921 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ) | |
26 | gcdmultiple 16128 | . . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ (abs‘𝑁) ∈ ℕ) → (𝑀 gcd (𝑀 · (abs‘𝑁))) = 𝑀) | |
27 | 25, 26 | sylan2 596 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑀 gcd (𝑀 · (abs‘𝑁))) = 𝑀) |
28 | 27 | anassrs 471 | . . 3 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → (𝑀 gcd (𝑀 · (abs‘𝑁))) = 𝑀) |
29 | 16, 24, 28 | 3eqtr3d 2787 | . 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → (𝑀 gcd (𝑀 · 𝑁)) = 𝑀) |
30 | mul01 11040 | . . . . . 6 ⊢ (𝑀 ∈ ℂ → (𝑀 · 0) = 0) | |
31 | 30 | oveq2d 7250 | . . . . 5 ⊢ (𝑀 ∈ ℂ → (𝑀 gcd (𝑀 · 0)) = (𝑀 gcd 0)) |
32 | 4, 31 | syl 17 | . . . 4 ⊢ (𝑀 ∈ ℕ → (𝑀 gcd (𝑀 · 0)) = (𝑀 gcd 0)) |
33 | 32 | adantr 484 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd (𝑀 · 0)) = (𝑀 gcd 0)) |
34 | nn0gcdid0 16112 | . . . . 5 ⊢ (𝑀 ∈ ℕ0 → (𝑀 gcd 0) = 𝑀) | |
35 | 9, 34 | syl 17 | . . . 4 ⊢ (𝑀 ∈ ℕ → (𝑀 gcd 0) = 𝑀) |
36 | 35 | adantr 484 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 0) = 𝑀) |
37 | 33, 36 | eqtrd 2779 | . 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd (𝑀 · 0)) = 𝑀) |
38 | 3, 29, 37 | pm2.61ne 3030 | 1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd (𝑀 · 𝑁)) = 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2943 ‘cfv 6400 (class class class)co 7234 ℂcc 10756 0cc0 10758 · cmul 10763 ℕcn 11859 ℕ0cn0 12119 ℤcz 12205 abscabs 14829 gcd cgcd 16085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10814 ax-resscn 10815 ax-1cn 10816 ax-icn 10817 ax-addcl 10818 ax-addrcl 10819 ax-mulcl 10820 ax-mulrcl 10821 ax-mulcom 10822 ax-addass 10823 ax-mulass 10824 ax-distr 10825 ax-i2m1 10826 ax-1ne0 10827 ax-1rid 10828 ax-rnegex 10829 ax-rrecex 10830 ax-cnre 10831 ax-pre-lttri 10832 ax-pre-lttrn 10833 ax-pre-ltadd 10834 ax-pre-mulgt0 10835 ax-pre-sup 10836 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-om 7666 df-2nd 7783 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-er 8414 df-en 8650 df-dom 8651 df-sdom 8652 df-sup 9087 df-inf 9088 df-pnf 10898 df-mnf 10899 df-xr 10900 df-ltxr 10901 df-le 10902 df-sub 11093 df-neg 11094 df-div 11519 df-nn 11860 df-2 11922 df-3 11923 df-n0 12120 df-z 12206 df-uz 12468 df-rp 12616 df-seq 13606 df-exp 13667 df-cj 14694 df-re 14695 df-im 14696 df-sqrt 14830 df-abs 14831 df-dvds 15848 df-gcd 16086 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |